Thermodynamics from a scaling Hamiltonian

TL;DR

This paper reviews nonextensive thermodynamics for long-range interacting systems and introduces a new scaled Hamiltonian that improves the understanding of their thermodynamic behavior, validated by Monte Carlo simulations.

Contribution

It proposes a novel Hamiltonian scaling method that explicitly accounts for system size and lattice symmetry, enhancing previous models of long-range interactions.

Findings

The new Hamiltonian scaling improves thermodynamic descriptions of long-range systems.

Monte Carlo simulations confirm the effectiveness of the scaled Hamiltonian.

The approach clarifies the role of system size and symmetry in nonextensive thermodynamics.

Abstract

There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance $r$ as $1/r^{\alpha}$ for $\alpha\geq0$. Here, we review old nonextensive scaling. In addition, we propose a new Hamiltonian scaled by $2\frac{(N/2)^{1-\alpha}-1}{1-\alpha}$ that explicitly includes symmetry of the lattice and dependence on the size, $N$, of the system. The new approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.